1. If \[ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = 2\frac{51}{79} \] then find the value of \(\sin\theta\).

2. Prove that \[ \cfrac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1} = \cfrac{1 + \sin\theta}{\cos\theta} \] If you'd like, I can help walk through the proof step-by-step as well. Ready when you are!

3. If \[ \frac{\sec \theta + \tan \theta}{\sec \theta - \tan \theta} = \frac{2 + \sqrt{3}}{2 - \sqrt{3}}, \] then what is the value of \( \theta \)?

(a) 60° (b) 30° (c) 45° (d) 90°

4. Prove that \[ \sqrt{\cfrac{1 + \cos \theta}{1 - \cos \theta}} = \csc \theta + \cot \theta \]

5. If \[ \tan \theta \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \] then find the value of \[ \sin(\theta - 15^\circ) \] given that \(0^\circ < \theta < 90^\circ\).

6. If \(\tan\theta = \frac{x}{y}\), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2-y^2}{x^2+y^2}\) (b) \(\cfrac{y^2-x^2}{x^2+y^2}\) (c) \(\cfrac{x^2+y^2}{y^2-x^2}\) (d) None of the above

7. If \( \tan\theta = \frac{x}{y} \), then find the value of \[ \frac{x \sin\theta - y \cos\theta}{x \sin\theta + y \cos\theta} \]

8. If \[ x \cot\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{3}\right) + \frac{3}{4} \sec^2\left(\frac{\pi}{4}\right) + 4 \sin\left(\frac{\pi}{6}\right) \] then find the value of \(x\).

9. Prove that \[ \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta}{\cos \theta} \]

10. Solve: \[ \tan^2 \frac{\pi}{4} \cdot \sin \frac{\pi}{x} \cdot \tan \frac{\pi}{6} \cdot \tan^2 \frac{\pi}{3} = 1\frac{1}{2} \]

11. If \[ \sin\theta + \cos\theta = \frac{7}{5} \quad \text{and} \quad \sin\theta \cos\theta = \frac{12}{25}, \] then show that \( \sin\theta = ? \)

12. If \( \sin\theta + \sin^2\theta = 1 \), then prove: \[ \cos^2\theta + \cos^4\theta = 1 \]

13. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

14. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \]

(a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

15. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

16. If \(x = \cfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \cfrac{x^2 + xy + y^2}{x^2 - xy + y^2} = \cfrac{12}{11} \]

17. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

18. If \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

19. If \[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3 \] then find the value of \[ \sin^4 \theta - \cos^4 \theta \]

20. If \[ x = \frac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}} \] then find the value of \[ \frac{x + \sqrt{3}}{x - \sqrt{3}} + \frac{x + \sqrt{5}}{x - \sqrt{5}} \]

21. Given: \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\) Prove that: \[ m^2 - n^2 = 4\sqrt{mn} \quad \text{where } 0^\circ < \theta < 90^\circ \]

22. Given: \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] Find the value of: \[ \frac{x^2 + 3xy + y^2}{x^2 - 3xy + y^2} \]

23. If \(x = \frac{\sqrt{3}}{2}\), then find the value of \[ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} - \sqrt{1 - x}} \]

24. If \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}? \]

(a) \(2\sqrt{3}\) (b) \(\cfrac{1}{\sqrt{3}}\) (c) \(\sqrt{3}\) (d) \(\sqrt{3}\)

25. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(xy = 1\), prove that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

26. Show that \[\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} - \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = 2\sqrt{15}\]

27. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x - \frac{1}{x} \]

28. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ y^2 + \frac{1}{y^2} \]

29. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ x^3 - \frac{1}{x^3} \]

30. If \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\), then find the value of \[ xy + \frac{1}{xy} \]